Pressurizing the fuel side with hydrogen might work in theory, but there are several practical complications:

- Hydrogen embrittlement. Exposure to hydrogen damages many kinds of metal. - Gases leak, especially gases with a low molecular weight, which is no big deal with helium, but can create an invisible fire with hydrogen. - Mistakes happen, and accidentally pressurizing the LOX tank with GH2 would be a disaster. - Using two pressurant gases instead of one adds complexity, which reduces operability, increases the number of things that can go wrong, and costs effort, time, and money. - Everybody uses helium pressurants; hardly anyone uses H2. This would make it a "science project", which adds R&D costs, and most importantly wastes time.

Helium is expensive but not expensive enough to justify switching half the vehicle to hydrogen.